At the end of the poem Sir Gawain and the Green Knight, what is the Green Knight's opinion of Gawain?

The short answer to your question is that the Green Knight
approves of Sir Gawain or he would be dead.  The challenge which begins the tale of
Sir Gawain and the Green Knight is a simple one--Sir Gawain will
allow someone to chop his head off today, but a year from now he must find the Green
Knight's home and allow his own head to be chopped off by the Green Knight.  Needless to
say, there were very few takers for this challenge--only two, in fact.  King Arthur and
the young but passionate Sir Gawain.

All went as planned,
and a year later Sir Gawain was in the home, though unbeknownst to him, of the Green
Knight.  Gawain did fulfill his obligations in an honorable way--almost.  In the deal
they made, Gawaindid withhold the scarf from his host, and he received a nick on his
neck as his punishment.  But that was it.  Rather than chop Gawain's head off, as was
his right, the Green Knight treated him like a man of honor.  He
said:

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"Therefore I bid thee, knight, come...and make
merry in thine house; ...I wish thee as well as any man on earth, by my faith, for thy
true dealing."

The Green
Knight admired him enough to invite him back to his home, though Gawain demurs, and
wishes him blessings on his journey.

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[S]o they embraced and kissed, and commended each
other to the Prince of Paradise, and parted right there, on the cold
ground.

This is the picture
of courtly behavior between two knights.  There is little doubt that Sir Gawain passed
the test of honor given him by the Green Knight and he was treated accordingly--with
respect and commendation.